To determine the linearity of a molecule or ion, we need to analyze its hybridization and the presence of lone pairs on the central atom. A linear structure typically arises from sp hybridization with no lone pairs on the central atom, or in cases where steric hindrance forces a linear arrangement.

The Valence Shell Electron Pair Repulsion (VSEPR) theory helps predict the geometry of molecules based on minimizing repulsion between electron pairs around the central atom. Hybridization describes the mixing of atomic orbitals to form new hybrid orbitals, which then determine the molecular geometry.

Let's analyze each option:

• A) NO₂⁺ (Nitronium ion): The central atom is Nitrogen. The total number of valence electrons is 5 (N) + 2 × 6 (O) - 1 (charge) = 16 electrons. The Lewis structure is O=N⁺=O. Nitrogen forms two double bonds and has no lone pairs. The steric number for nitrogen is 2 (two bond pairs + zero lone pairs). This corresponds to sp hybridization, leading to a linear geometry.
• B) NCO⁻ (Cyanate ion): The central atom can be N or C. If C is central: N⁻=C=O. Total valence electrons = 5 (N) + 4 (C) + 6 (O) + 1 (charge) = 16 electrons. Carbon forms two double bonds and has no lone pairs. Steric number = 2, sp hybridization, linear geometry. If N is central: O=N=C⁻. This is less stable due to negative charge on carbon. However, the question asks for 'the linear structure assumed by', implying only one correct option. Both NCO⁻ and CS₂ are linear. Let's re-evaluate the question. The question asks for 'the linear structure assumed by', implying only one correct option. Let's check the other options carefully.
• C) SnCl₂ (Tin(II) chloride): The central atom is Tin (Sn). Total valence electrons = 4 (Sn) + 2 × 7 (Cl) = 18 electrons. Sn forms two single bonds and has one lone pair. The steric number for Sn is 3 (two bond pairs + one lone pair). This corresponds to sp² hybridization, leading to a bent or V-shaped geometry.
• D) CS₂ (Carbon disulfide): The central atom is Carbon. Total valence electrons = 4 (C) + 2 × 6 (S) = 16 electrons. Carbon forms two double bonds with sulfur atoms and has no lone pairs. The steric number for carbon is 2 (two bond pairs + zero lone pairs). This corresponds to sp hybridization, leading to a linear geometry.

Upon re-evaluation, both NO₂⁺ and CS₂ are linear. NCO⁻ is also linear. This suggests there might be an ambiguity in the question or options provided, as multiple options exhibit a linear structure. However, in typical MCQ scenarios, if multiple options are linear, there might be a subtle distinction or a most common/representative example. Given the options, NO₂⁺ is a classic example of a linear ion. If only one answer is to be chosen, and assuming the question expects a single best fit, let's stick to the provided correct answer A.

Let's re-examine the options assuming there's only one correct answer. All three (NO₂⁺, NCO⁻, CS₂) are indeed linear based on VSEPR and hybridization. If this is a single-choice question, there might be an error in the question or options, or a specific context is missing. However, if forced to choose based on the provided answer, we proceed with the analysis for A.

For NO₂⁺:

1. Central Atom: Nitrogen (N)
2. Valence Electrons: N (5) + 2 × O (6) - 1 (charge) = 5 + 12 - 1 = 16 electrons.
3. Lewis Structure: O=N⁺=O. Nitrogen forms two double bonds.
4. Steric Number: 2 (2 bond pairs, 0 lone pairs).
5. Hybridization: sp.
6. Geometry: Linear.

Option Analysis:

• A) NO₂⁺: Linear due to sp hybridization on the central nitrogen atom with no lone pairs.
• B) NCO⁻: Linear due to sp hybridization on the central carbon atom with no lone pairs.
• C) SnCl₂: Bent (V-shaped) due to sp² hybridization on the central tin atom with one lone pair.
• D) CS₂: Linear due to sp hybridization on the central carbon atom with no lone pairs.

Given that the provided correct answer is A, and acknowledging that B and D are also linear, the question might be flawed or seeking a specific context not provided. However, based on the standard VSEPR theory, NO₂⁺ is definitively linear.

Correct Answer: (A)